AUM block mean labeling assigns labels to the blocks of a graph by averaging the labels of the vertices and edges. In this paper, a new variant of AUM block labeling using Gaussian integers, namely AUM Gaussian block labeling is introduced. An ordering of Gaussian integers in lower triangular matrix form, in which the Gaussian integers lie in the first quadrant, is used to define AUM Gaussian block labeling to assign the labels to the blocks of a graph. Further, AUM Gaussian block labeling is discussed for some unicyclic graphs in this work. An application of AUM Gaussian block labeling in the encryption and decryption processes of coding theory by integrating Euler’s function and the norm of a Gaussian integer is also presented.